# The General Theory of Set Valued Semidynamical Systems

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## Abstract

A set-valued mapping or correspondence, ө:
where x ∈

**r**→**r**is defined along with a family of correspondences, parameterized by t, in the following manner:$$ {{\theta }^{0}}(X): = X $$

(3.1)

$$ {{\theta }^{t}}(X): = \theta ({{\theta }^{{t - 1}}}(X))\quad \forall t\varepsilon {{{\rm N}}_{ + }} $$

(3.2)

**r**:**Q**×**P**. Hence Θ^{t}is the formal solution of the set valued difference equation X_{t+1}. Since Θ is a USC, Θ^{t}is a USC for all t ∈**N**_{+}when Θ(X) =U_{X∈X}Θ{X}. The family of correspondences {Θ^{t}|t ∈**N**_{+}forms a semigroup of operators, for Θ^{0}is an identity operator, and for all r, s ∈**N**_{+}and × ∈*r*. Θ^{r}(Θ^{s}(X}}=Θ^{r+s}. The value of Θ^{t}(X) is the set in which motion starting in the set X will be located after the duration of length t. This semigroup is analogous to the single valued solution of a difference or differential equation in single valued systems where, for each t, an initial condition corresponding to X is mapped to some other position in the state space. For such single valued systems, {Θ^{t}} is a set of continuous mappings.## Keywords

Accumulation Point Global Attractor Strong Attractor Closed Mapping Weak Attractor
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 1978